\chapter{Introduction}

\faust (\textit{Functional Audio Stream}) is a functional programming language specifically designed for real-time signal processing and synthesis.  \faust targets high-performance signal processing applications and audio plug-ins for a variety of platforms and standards. 

\section{Design Principles} 

Various principles have guided the design of \faust :

\begin{itemize}

\item \faust is a \textit{specification language}. It aims at providing an adequate notation to describe \textit{signal processors} from a mathematical point of view. \faust is, as much as possible, free from implementation details. 

\item \faust programs are fully compiled, not interpreted. The compiler translates \faust programs into equivalent C++ programs taking care of generating the most efficient code. The result can generally compete with, and sometimes even outperform, C++ code written by seasoned programmers. 

\item The generated code works at the sample level. It is therefore suited to implement low-level DSP functions like recursive filters. Moreover the code can be easily embedded. It is self-contained and doesn't depend of any DSP library or runtime system. It has a very deterministic behavior and a constant memory footprint. 

\item The semantic of \faust is simple and well defined. This is not just of academic interest. It allows the \faust compiler to be \emph{semantically driven}. Instead of compiling a program literally, it compiles the mathematical function it denotes. This feature is useful for example to promote components reuse while preserving optimal performance.  

\item \faust is a textual language but nevertheless block-diagram oriented. It actually combines two approaches: \textit{functional programming} and \textit{algebraic block-diagrams}. The key idea is to view block-diagram construction as function composition. For that purpose, \faust relies on a \emph{block-diagram algebra} of five composition operations (\lstinline': , ~ <: :>').

\item Thanks to the notion of \textit{architecture}, \faust programs can be easily deployed on a large variety of audio platforms and plugin formats without any change to the \faust code.

\end{itemize}

\section{Signal Processor Semantic}
A \faust program describes a \emph{signal processor}. 
The role of a \textit{signal processor} is to transforms a (possibly empty) group  of \emph{input signals} in order to produce a (possibly empty) group of \emph{output signals}. 
Most audio equipments can be modeled as \emph{signal processors}. 
They have audio inputs, audio outputs as well as control signals interfaced with sliders, knobs, vu-meters, etc. 

More precisely :

\begin{itemize}

\item A \emph{signal} $s$ is a discrete function of time $s:\mathbb{Z}\rightarrow\mathbb{R}$.
\marginpar{\faust considers two type of signals: \emph{integer signals} ($s:\mathbb{Z}\rightarrow\mathbb{Z}$) and \emph{floating point signals} ($s:\mathbb{Z}\rightarrow\mathbb{Q}$) Exchanges with the outside world are, by convention, made using floating point signals. The full range is represented by sample values between -1.0 and +1.0.}
The value of a signal $s$ at time $t$ is written $s(t)$. The values of signals are usually needed starting from time $0$. But to take into account \emph{delay operations}, negative times are possible and are always mapped to zeros. Therefore for any \faust signal $s$ we have $\forall t<0, s(t)=0$. In operational terms this corresponds to assuming that all delay lines are signals initialized with $0$s.
 
\item The set of all possible  signals is $\mathbb{S}=\mathbb{Z}\rightarrow\mathbb{R}$.

\item A group of $n$ signals (a \emph{n}-tuple of signals) is written
$(s_{1},\ldots,s_{n})\in \mathbb{S}^{n}$.
The \emph{empty tuple}, single element of $\mathbb{S}^{0}$ is notated  $()$.

\item A \emph{signal processors} $p$, is a function from
\emph{n}-tuples of signals to \emph{m}-tuples of signals
$p:\mathbb{S}^{n}\rightarrow\mathbb{S}^{m}$. The set $\mathbb{P}=\bigcup_{n,m}\mathbb{S}^{n}\rightarrow\mathbb{S}^{m}$ is the
set of all possible signal processors.

\end{itemize}

As an example, let's express the semantic of the \faust primitive \lstinline'+'. Like any \faust expression, it is a signal processor. Its signature is $\mathbb{S}^{2}\rightarrow\mathbb{S}$. It takes two input signals $X_0$ and $X_1$ and produce an output signal $Y$ such that $Y(t) = X_0(t)+X_1(t)$. 

Numbers are signal processors too. For example the number $3$ has signature  $\mathbb{S}^{0}\rightarrow\mathbb{S}$.  It takes no input signals and produce an output signal $Y$ such that $Y(t) = 3$. 


